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Summaries or graphs?

Herb Edelstein from Two Crows consulting introduced me to this neat example showing how graphs are much more revealing than summary statistics. This is an age-old example by Anscombe (1973). I will show a slightly updated version of Anscombe’s example, by Basset et al. (1986):We have four datasets, each containing 11 pairs of X and Y measurements. All four datasets have the same X variable, and only differ on the Y values.

Here are the summary statistics for each of the four Y variables (A, B, C, D):

A

B

C

D

Average

20.95

20.95

20.95

20.95

Std

1.495794

1.495794

1.495794

1.495794

That’s right – the mean and standard deviations are all identical. Now let’s go one step further and fit the four simple linear regression models Y= a + bX + noise. Remember, the X is the same in all four datasets. Here is the output for the first dataset:

Regression Statistics

Multiple R

0.620844098

R Square

0.385447394

Adjusted R Square

0.317163771

Standard Error

1.236033081

Observations

11

Coefficients

Standard Error

t Stat

P-value

Intercept

18.43

1.12422813

16.39347

5.2E-08

slope

0.28

0.11785113

2.375879

0.041507

Guess what? The other three regression outputs are identical!

So are the four Y variables identical???

Well, here is the answer:To top it off, Basset included one more dataset that has the exact same summary stats and regression estimates. Here is the scatterplot:

[You can find all the data for both Anscombe’s and Basset et al.’s examples here]